Removing Anti-Aliasing from shadows in the BGE
This can be solved relatively easily, but you may have to make minor changes elsewhere to keep the crisp shadow edge as pictured here, only without the anti-aliasing

This is the type of shadow that you have now.



and I assume this is closer to what you want.



You can achieve this simply be changing the shadow casting setting on your shadow emitting lamps from Simple to Variance



of course, now you can tweak the settings to restore the crisp shadow edge. I found that by squashing the Bias, the shadow looked much better.



You will need to make all your shadow lamps have the same settings for the final result to look good. It is usually a good idea to only have a  few shadow lamps, one main lamp for everything, and then extra spotlights as needed. A better approach is baking on the shadows for the environment, and then use a spotlight parented to the character to give your character dynamic shadows.